Last week's local elections, which saw disastrous results for Conservatives and disappointing losses for Labour, have increased pressure on both sides for a breakthrough in the cross-party talks, which have run on for over a month without producing agreement.
Lidington's admission also suggests that cross-party talks between May's team and the Labour party will not bear fruit any time soon.
Brady, the chairman of the so-called 1922 committee of Tory MPs, said she agreed to address its executive members next week on the question of her future.
Nearly three years since Britain voted to leave the European Union, there is little clarity about how, when or even if Brexit will happen.
However, Downing Street has insisted Mrs May will only leave once a Brexit deal has been passed.
Clearly frustrated by the attack on her leadership, May expressed her regret at the significant defeat inflicted upon the Conservative party by voters last week, which has been broadly interpreted in Westminster as a damning indictment of the party's inability to deliver Brexit, but refused to take the blame.
"If they don't produce a manifesto, the Prime Minister's word will be the word of the Conservative Party, and I have to make it clear that I would not vote for the Prime Minister's policy and I think a very large number of people feel exactly the same".
Previously loyal Tory donors were "asking themselves the question 'what is the Conservative Party for, what objective does it actually serve?'". In addition, local Conservative associations have called for an extraordinary general meeting, to be held on 15 June, which is expected to see a non-binding vote of no confidence in May.
But former Labour MP Caroline Flint, despite supporting Labour pushing for a Brexit deal with the Tories, said there was not a majority among MPs for a second referendum.
"It would be much easier and I think the European elections would be much easier if she did set out her own timetable to go but it is up to her". Pro-EU Labour voters could turn to Change UK, another new party, or support the Liberal Democrats.
"This week is meant to be the "deadline" for these talks - talking of cut-off points, David Lidington has claimed July 2 as the latest government deadline for passing a Brexit deal - so an outcome, positive or negative, could be confirmed in the coming days".
May has pinned hopes of securing Parliament's support for a Brexit deal on reaching a compromise with the opposition Labour Party.
"We are fighting the European elections here in the U.K. Why?" She stressed that she was aware that the MPs didn't want her to remain at the helm during the next phase of Brexit talks, and she didn't want to "stand in the way of that".